10.2 Resolution
Definition 10.2.1. The resolution of a fractional factorial design is the length (number of letters) of the shortest word in its defining relation, written as a Roman numeral. In particular:
- (i)
- in a design of resolution III, no main effect is aliased with another main effect, but main effects are aliased with two-factor interactions, and two-factor interactions may be aliased with each other;
- (ii)
- in a design of resolution IV, no main effect is aliased with another main effect or with a two-factor interaction, but two-factor interactions are aliased with each other;
- (iii)
- in a design of resolution V, no main effect or two-factor interaction is aliased with another main effect or two-factor interaction.
The connection with word length is immediate from Theorem 10.1.2: an effect with \(p\) letters is aliased with an effect of at least \(R - p\) letters when the shortest word has \(R\) letters. The \(2^{3-1}\) design with \(I = ABC\) is \(2^{3-1}_{\mathrm {III}}\), and the \(2^{4-1}\) with \(I = ABCD\) is \(2^{4-1}_{\mathrm {IV}}\).
Proposition 10.2.2. The one-half fraction of the \(2^k\) design of highest resolution has defining relation \(I = ABC\cdots K\), the interaction of all \(k\) factors, and is of resolution \(k\). It is constructed by writing down the full \(2^{k-1}\) design in the first \(k - 1\) factors, and setting the levels of the \(k\)th factor equal to the product of the columns of the other \(k - 1\) (or minus that product, for the alternate fraction).
Proof. A half fraction has a single word in its defining relation, and the resolution is that word’s length, at most \(k\), with equality only for the word containing every letter. Setting \(K = AB\cdots J\) makes \(ABC\cdots K = (AB\cdots J)^2 = I\) on every run, so the runs constructed are exactly those with \(+\) in the \(ABC\cdots K\) column, and there are \(2^{k-1}\) of them, all distinct. □
A useful consequence of resolution is the projection property: a design of resolution \(R\) contains a full factorial (possibly replicated) in any \(R - 1\) of its factors. If only a few factors turn out to be active, the data can be analysed as a full factorial in those factors.
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